Question: The lifespans of bears in a particular zoo are normally distributed. The average bear lives $34$ years; the standard deviation is $8$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a bear living between $42$ and $50$ years.
Solution: $34$ $26$ $42$ $18$ $50$ $10$ $58$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $34$ years. We know the standard deviation is $8$ years, so one standard deviation below the mean is $26$ years and one standard deviation above the mean is $42$ years. Two standard deviations below the mean is $18$ years and two standard deviations above the mean is $50$ years. Three standard deviations below the mean is $10$ years and three standard deviations above the mean is $58$ years. We are interested in the probability of a bear living between $42$ and $50$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the bears will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the bears will have lifespans within 1 standard deviation of the mean. The probability of a particular bear living between $42$ and $50$ years is $\color{orange}{13.5\%}$.